A372530
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x)) ).
Original entry on oeis.org
1, 1, 3, 9, 33, 125, 501, 2065, 8739, 37685, 165107, 732681, 3286679, 14878885, 67889851, 311896993, 1441536321, 6698017445, 31269529601, 146601334841, 689945263873, 3258334336349, 15436401872405, 73341269533009, 349381321611505, 1668434132560765, 7985390073708765
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 33*x^5 + 125*x^6 + 501*x^7 + 2065*x^8 + 8739*x^9 + 37685*x^10 + 165107*x^11 + 732681*x^12 + ...
where A( x*A(x)/(1 - A(x)) ) = A(x)^2.
RELATED SERIES.
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(2^n)) = x - x^2 - x^3 + x^4 - x^5 + x^6 + x^7 - x^8 - x^9 + x^10 + x^11 - x^12 + x^13 - x^14 - x^15 + x^16 + ... + (-1)^A010060(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x)) * (1 - A(x)^2) * (1 - A(x)^4) * (1 - A(x)^8) * (1 - A(x)^16) * ... * (1 - A(x)^(2^n)) * ...
SPECIFIC VALUES.
A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(2^n)) = 0.195062471888103139123433255203480726664398592...
A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(2^n)) = 0.175091932719784804433277263483089433821043251...
A(1/6) = 0.2285942310240955503097133963953487564542629539800372181...
A(1/7) = 0.1803372891149269875688065840927292319030238580575714990...
A(1/8) = 0.1506715662175837437127190414569072051853697889895576799...
A(1/6)^2 = A(t) at t = (1/6)*A(1/6)/(1 - A(1/6)) = 0.0493891023845...
A(1/7)^2 = A(t) at t = (1/7)*A(1/7)/(1 - A(1/7)) = 0.0314305744685...
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{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0); F=Ser(A);
A[#A] = polcoeff( subst(F,x, x*F/(1 - F) ) - F^2, #A) ); H=A; A[n+1]}
for(n=1,30, print1(a(n),", "))
A371716
Expansion of g.f. A(x) satisfies A( x*A(x)^3 + x*A(x)^4 ) = A(x)^4.
Original entry on oeis.org
1, 1, 1, 1, 2, 7, 22, 57, 131, 298, 738, 2003, 5600, 15380, 41224, 109769, 296010, 813333, 2261818, 6307070, 17560050, 48877852, 136457322, 382803675, 1078562370, 3047295816, 8623046992, 24432992884, 69345396556, 197211214852, 561975160288, 1604186098089, 4585779820379
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 7*x^6 + 22*x^7 + 57*x^8 + 131*x^9 + 298*x^10 + 738*x^11 + 2003*x^12 + 5600*x^13 + 15380*x^14 + ...
where A( x*A(x)^3*(1 + A(x)) ) = A(x)^4.
RELATED SERIES.
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) = x/((1+x)*(1+x^4)*(1+x^16)*(1+x^64)*(1+x^256)*(1+x^1024)*...) = x - x^2 + x^3 - x^4 + x^9 - x^10 + x^11 - x^12 + x^33 - x^34 + ...
We can show that g.f. A(x) = A( x*A(x)^3*(1 + A(x)) )^(1/4) satisfies
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(4^n))
by substituting x*A(x)^3*(1 + A(x)) for x in (4) to obtain
A(x)^4 = x * A(x)^3*(1 + A(x)) * Product_{n>=1} (1 + A(x)^(4^n))
which is equivalent to formula (4).
SPECIFIC VALUES.
A(1/3) = 0.6209428791888803994421374991623399343094...
A(1/4) = 0.3392462304609640143453810140211726768116...
A(1/5) = 0.2512464727722296135954631316870173555867...
A(t) = 1/2 and A(t*3/16) = 1/16 at t = 0.31372070319804379323613829910755157...
A(t) = 1/3 and A(t*4/81) = 1/81 at t = 0.24695121377537689193140239461709572...
A(t) = 1/4 and A(t*5/256) = 1/256 at t = 0.199221789836883544932674834867379...
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/* Using series reversion of x/Product_{n>=0} (1 + x^(4^n)) */
{a(n) = my(A); A = serreverse( x/prod(k=0, ceil(log(n)/log(4)), (1 + x^(4^k) +x*O(x^n)) ) ); polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
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/* Using A(x)^4 = A( x*A(x)^3 + x*A(x)^4 ) */
{a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^3 + x*F^4 ) - F^4, #A+3) ); A[n]}
for(n=1, 35, print1(a(n), ", "))
A376230
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).
Original entry on oeis.org
1, 1, 3, 8, 28, 98, 370, 1423, 5643, 22753, 93299, 387324, 1625768, 6886156, 29399430, 126377000, 546527682, 2376094442, 10379414436, 45532904886, 200511864604, 886055084460, 3927826810396, 17462128520246, 77838085223570, 347813389031746, 1557683052973482, 6990670698115144
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 28*x^5 + 98*x^6 + 370*x^7 + 1423*x^8 + 5643*x^9 + 22753*x^10 + 93299*x^11 + 387324*x^12 + ...
where A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).
RELATED SERIES.
Let B(x) be the g.f. of Stern's diatomic series (A002487):
B(x) = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + 4*x^9 + 3*x^10 + 5*x^11 + 2*x^12 + 5*x^13 + 3*x^14 + 4*x^15 + x^16 + ...
then A(x)^2 = x * B( A(x) ) and A( x^2/B(x) ) = x.
Other related series begin as follows.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 81*x^6 + 300*x^7 + 1168*x^8 + 4622*x^9 + 18704*x^10 + 76738*x^11 + 319054*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 43*x^6 + 168*x^7 + 657*x^8 + 2649*x^9 + 10803*x^10 + 44733*x^11 + 187130*x^12 + ...
SPECIFIC VALUES.
A(t) = 2/5 at t = 0.21059927171685797509442589615351221149308548505349232763302...
A(t) = 1/3 at t = 0.20284350479378267499481171039846064829566135845385597426623...
A(t) = 1/4 at t = 0.17791461653470954766421766118399907657065676113208935616640...
A(t) = 1/5 at t = 0.15460046705113920070261079885331294477343970681952336915318...
A(1/5) = 0.31995483821441278259163540295892975411660207078522958569307...
A(1/6) = 0.22418805161328879302723377308422267982113532037722470920139...
A(1/7) = 0.17887744157158359437462802053243127220002079340556427992475...
A(1/8) = 0.15001315387877904231502214457445835910409883718703588438530...
A(1/10) = 0.11423875178408085947774841103426888472717517658954942399943...
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/* Series_Reversion(x / Product_{n>=0} (1 + x^(2^n) + x^(2^(n+1))) ) */
{a(n) = my(A = serreverse(x / prod(k=0, #binary(n), (1 + x^(2^k) + x^(2^(k+1))) +x*O(x^n)))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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/* A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ) */
{a(n) = my(A=[1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A);
A[#A] = -polcoeff( F^2 - subst(F, x, x*F + x*F^2 + x*F^3), #A+1) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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